The conjecture is equivalent to the following:
Among any $2n$ consecutive integers below $n^2+2n$, there is at least one prime greater than $n$.
Indeed: if your conjecture is true and none of the numbers in the sequence would be prime, then at least one number is the product of at least two primes $>n$, so is at least $(n+1)^2$, contradiction.
In particular it says:
There is at least one prime among $n^2,\ldots,n^2+2n-1$
which (because $n^2$ and $n^2+2n$ are never prime) is the same as
There is at least one prime among $n^2+1,\ldots,n^2+2n$
which is Legendre's conjecture and is unsolved.